Abstract
Interactions between the constituents of a condensed matter system can drive it through a plethora of different phases due to manybody effects. A prominent platform for it is a dilute twodimensional electron system in a magnetic field, which evolves intricately through various gaseous, liquid and solid phases governed by Coulomb interaction. Here we report on the experimental observation of a phase transition between the composite fermion liquid and adjacent magnetic field induced phase with a character of Wigner solid. The experiments are performed in the lowest Landau level of a MgZnO/ZnO twodimensional electron system with attributes of both a liquid and a solid. An inplane magnetic field component applied on top of the perpendicular magnetic field extends the Wignerlike phase further into the composite fermion liquid phase region. Our observations indicate the direct competition between a composite fermion liquid and a Wigner solid formed either by electrons or composite fermions.
Introduction
A magnetic field B applied perpendicularly to a twodimensional charge carrier system modifies its density of states and places the charge carriers on a ladder of discrete Landau levels (LL). The Coulomb interaction between the charged particles acting on the magnetic length scale \(l_{\mathrm{B}} = \sqrt {\hbar /eB}\) can be tuned by varying the magnetic field strength. Thereby, the high mobility carriers evolve through the various correlation phases^{1}. When the electrons occupy half of available states in the lowest LL, e.g., filling factor ν = 1/2, the electrons prefer to reduce their interaction by virtue of capturing two magnetic flux quanta resulting in the emergence of new particles, called composite fermions (CF)^{2,3}. These particles form a Fermi surface at ν = 1/2 and move in an effective field B_{eff} = B − B_{ν = 1/2} (Fig. 1, middle panel) giving rise to magnetoresistance oscillations. At even lower filling factors a Wigner solid, a crystalline phase of charged particles (electrons or CF) driven by the repulsive Coulomb force and yet another manifestation of manybody correlations, emerges as a ground state of the electron system (Fig. 1).
Being in the lowest Landau level (LL) the electron system experiences competition between the composite fermion liquid phase and the magnetic field induced Wigner solid phase, which manifests as a large magnetoresistance peak around or below ν = 1/3^{4,6}. A liquidsolid transition may follow the Kosterlitz–Thouless model, whereas the particles can form a hexatic phase characterized by bondoriented nearestneighbor ordering^{7,8,9,10,11}. An intermediate phase of the liquidsolid transition may also take the form of microemulsion phases associated with a liquid crystalline phase^{11,12,13}. Departing from the liquid phase of CF at ν = 1/2, a formation of both a composite fermion Wigner solid and phase transition to intermediate phases may appear feasible. The idea of realizing a composite fermion Wigner solid was put forward in a number of theoretical works^{14,15,16,17,18,19}. Recent experiments focusing on GaAsbased 2DES have been gradually accumulating evidence pointing towards the realization of CF Wigner solid^{20,21,22,23,24}. Intuitively the CF crystal is stabilized when the CF of nearby liquid states release two of their vortices to stabilize the crystal, whereas the undressed particles retain their energetically favorable correlations^{18}. Thus a twoflux CF crystal borders the fourflux composite fermion liquid phase, whereas an electron crystal phase is embedded in twoflux composite fermion liquid and forms between filling factors ν = 1/3 and ν = 2/5 for a high enough LL mixing^{19}. Thus the transition between the liquid and the solid can be highly nontrivial and is realized in the lowest LL of a twodimensional charge carrier system by the transformation of the underlying particle type.
Here, we study the magnetotransport in a ZnO heterostructure (see: Methods) in the magnetic field region between the CF liquid phase formed at ν = 1/2 and the high resistivity phase appearing at higher field and exhibiting attributes of a Wigner solid^{25,26}. LL mixing, the ratio between electron–electron interaction energy and the cyclotron energy, is 4.2 at ν = 1/2 in this heterostructure and the magnetotransport in the region of interest features a character of both CF liquid and crystalline phase. The presence of such a region with the interlaced character highlights a nontrivial nature of phase transition, the details of which can further be masked by the inhomogeneous potential landscape arising from inevitable crystallographic disorder. The transition between the two phases can be tuned by the application of an inplane magnetic field. As a result of the phase intermixture, the state at filling factor ν = 1/2 can be formed by a composite fermion liquid and some intermediate state arising in the course of liquidsolid transition. Owning to a simple band structure of ZnO as compared to ptype GaAs, this oxide material is an attractive platform to access the competition between liquid and solid phases in the fractional quantum Hall regime.
Results
Experiment in perpendicular magnetic field
Figure 2 shows a full scan of the magnetotransport from 0 T to 33 T applied perpendicular to the 2DES plane. Several fractional quantum Hall states are observed around ν = 3/2, consistent with previous results^{26} and, in addition, developing minima are observed at ν = 9/5, 12/7, 9/7, and 6/5. Furthermore, up to six fractional quantum Hall states are observed on both sides of ν = 1/2. Close inspection of the transport around ν = 1/2 reveals a distinct asymmetry; R_{xx} maxima between fractional quantum Hall states for ν < 1/2 are much larger than those for ν > 1/2. This increase becomes increasingly dramatic between ν = 2/5, 1/3, and 2/7. Such a high resistance phase between ν = 1/3 and 2/7 has also been observed in GaAs, and was interpreted as the electron Wigner solid pinned by disorder. Two mechanisms have been identified for the appearance of the Wigner solid around these filling factors: one is a large LL mixing, which modifies the ground state energies of fractional quantum Hall states and Wigner solid;^{4,5,19} the other is shortrange disorder^{4,27,28}. Both mechanisms are more pronounced in ZnOheterostructures than in GaAs^{25,29}.
In Fig. 2 the apparent high resistance phases are colored and marked as HRP1, HRP2, and HRP3. On the basis of Wigner solid studies in other materials system, the characteristics of HRP1, HRP2, and HRP3 are typical of the Wigner solid. The temperature dependence of R_{xx} for HRP1, HRP2, and HRP3 resembles the melting of the Wigner solid (Supplementary Fig. 1a). The nonlinear currentvoltage characteristics are associated with the depinning of the Wigner solid from the disorder, when a certain threshold force is exceeded, and its subsequent sliding along the disorder landscape (Supplementary Fig. 1b). Thus, the trace of a Wigner solid formation appears already in HRP1 between ν = 3/7 and ν = 2/5, whereas a larger R_{xx} and I–V nonlinearity at HRP2 and HRP3 indicate a stronger pronounced Wigner solid character.
A nonlinear I–V characteristics can also appear in structures with domain boundaries originating from the sample inhomogeneity. To reduce the effect of sample inhomogeneity the structures were being rotated during the molecular epitaxial growth and we used a piece of structure next to the center of the ZnO substrate to ensure a higher sample homogeneity^{30}. Although we cannot exclude the effect of sample inhomogeneity on the sample characteristics, our structure demonstrates magnetotransport of quality comparable with the best GaAs structures. Furthermore, as we will demonstrate later in the paper, the effect of inplane magnetic field on both the high resistive phases and the transport around ν = 1/2 has a largely intrinsic origin.
HRP1 represents an interesting region. While it shows the features of an emerging Wigner solid, at the same time it can be comprised as a part of the R_{xx} oscillations caused by the CFs’ orbital motion in B_{eff}, and therefore can also be attributed to the liquid phase. We note that the R_{xx} amplitude at HRP2 shoots up so high that it can hardly be associated with a gradual increase of the Shibnikovde Haas oscillations amplitude with an increasing magnetic field. The estimate of the CF mass can be valuable to characterize the CF liquid phase and to elucidate the characteristics of region HRP1. Thus, we now analyze the CF mass m_{CF} around ν = 1/2 from the temperature dependence of the R_{xx} oscillation amplitude by using the Lifshitz–Kosevitch formalism (Supplementary Note 2). Figure 2b displays m_{CF} around ν = 1/2, which extends the linear dependence of m_{CF} on B (Supplementary Fig. 3) to higher field as m_{CF}/m_{0} = 0.047B/T^{25}. More noticeable is the increase of m_{CF} over the linear trend when the 2DES approaches the high resistivity phase HRP1.
The increase of the composite fermion mass is the signature of an enhanced electron–electron interaction, effectively strengthened by the reduction of kinetic energy, which is brought about by the electron localization by the magnetic field and/or disorder. An enhanced mass can also possibly be anticipated when CFs become more inert due to their localization. Both scenarios for mass increase signal the formation of a solid state, which is consistent with observing the traces of the Wigner solid character at HRP1, and the more strongly pronounced Wigner solidlike character at HRP2. Our transport experiment cannot distinguish whether the high resistance phase HRP2 is formed by the electron or composite fermion. Following the recent theory^{19} HRP2 should correspond to an electron crystal. Then the transport characteristics described above can be treated as experimental attributes of the intricate phase transition between the composite fermion liquid phase and electron solid. HRP3 can potentially be attributed to a twoflux composite fermion Wigner crystal, as it forms in the LL filling factors attributed to a fourflux composite fermion liquid.
Experiment in tilted magnetic field
The transport properties discussed above change dramatically when the sample is rotated in the magnetic field, that is, when an additional field component is applied parallel to the 2DES. Since the electron spin susceptibility for this structure is about 2, the opposite spin orientation branch of the lowest LL lies energetically high and is not populated^{31}. Thus the spin effects are not anticipated to play a role for the discussion below. Figure 3a depicts R_{xx} traces at several sample orientations θ obtained at base temperature and shows the asymmetrical impact of the inplane field on the transport for ν < 1/2 and ν > 1/2 (θ is the tilt angle between the normal of the 2DES plane and the magnetic field direction). Firstly, one notices that R_{xx} of HRP1, HRP2, and HRP3 increases gradually with an increasing θ, while R_{xx} minima at ν = 3/7, 2/5, and 1/3 do not change significantly. Thus the insulating phase becomes more pronounced by applying an inplane field. The temperature dependence of R_{xx} maxima of the high resistivity phases is depicted for three representative θs in Fig. 3b. Furthermore, R_{xx} around ν = 1/2 gains a background, which becomes larger with the increasing θ. Since magnetotransport experiments in GaAs demonstrate the extension of the tail of the insulating phase into the ν = 1/2 region with an increasing θ^{32,33}, we may also suppose that the background forming around ν = 1/2 has the same origin and is associated with the development of the insulating phase growing towards ν = 1/2 and above.
Discussion
A strong inplane field couples to the orbital motion and enhances the intersubband scattering rate^{34}. This will result in an additional LL broadening and should be particularly effective in our ZnO heterostructure, since the subband separation in the confinement potential is of the order of 5 meV and the electron cyclotron energy reaches 3.8 meV at 10 T^{35}. The scenario of scattering rate enhancement at high θ is not consistent with our experimental data. Indeed, for all tilt angles θ the R_{xx} oscillations are not damped but rather persist on top of an increasing background. Both the field where R_{xx} oscillations onset and the number of R_{xx} oscillations do not change with an increasing inplane field strength, whereas an anticipated broader LL ought to smear out the oscillations and to increase the field where R_{xx} oscillations set on.
Furthermore, we analyze the temperature dependence of the R_{xx} oscillation amplitude and estimate m_{CF} around ν = 1/2 for several θs (Supplementary Note 2). The inset of Fig. 3 depicts the result of this analysis. For ν > 1/2, m_{CF} does not show any noticeable change, but it shows a pronounced field and tilt angle dependence for ν < 1/2, that is, for a given perpendicular magnetic field m_{CF} is heavier at a larger tilt angle. The quantum scattering time traced as a fitting parameter in the estimation of m_{CF} does not show a tilt angle dependence around ν = 1/2(Supplementary Fig. 5). This strengthens our concept of negligible change of intersubband scattering rate due to the application of inplane magnetic field. The estimation of scattering time becomes more uncertain at approaching the HRP1 phase. This points to an inadequate description of high resistivity phases with Lifschitz–Kosevitch approximation, possibly supporting our point that the HRP1 phase has a mixed character of state of matter. Therefore the experimental finding implies that the change in magnetotransport caused by the inplane field application has an intrinsic origin.
The application of an inplane field can significantly disturb the electron/CF Fermi contour making it elliptic shaped^{36,37,38,39,40}. The effective particle mass is now given by \(m \ast = \sqrt {m_t \times m_l}\), where m_{t} and m_{l} are the masses along two prime axises of the ellipse. Such anisotropy might be responsible for the CF mass enhancement beyond the mass estimated at zero tilt angle. An enhanced mass can also signal that the electron–electron interaction becomes stronger, which brings the system closer to the condition to form a crystalline phase.
In order to further address the inplane field induced stabilization of the Wigner solidlike phase we now analyze how much the inplane field squeezes the electron wavefunction width, as it effectively enhances the Coulomb interaction and can affect the transport properties^{41,42,43}. In zero inplane field the wavefunction width of the heterostructure is about 10 nm wide^{31,35}. At θ = 50° it is squeezed down to 2.6 nm at B_{⊥} = 12 T, representing the region ν > 1/2, and down to 2.2 nm at B_{⊥} = 17 T, representing the region ν < 1/2 (Supplementary Note 3). Since the wavefunction width reduces significantly with inplane field on both sides of ν = 1/2 compared with zero inplane field, the Coulomb interaction should also be commensurately enhanced around ν = 1/2. Nonetheless, m_{CF} defined by the interaction effects remains almost unchanged for ν > 1/2 and no large effect of inplane field on transport characteristics is seen in this region. Consequently, the increase of m_{CF} for ν < 1/2 is not mainly caused by the reduced wavefunction width. However, a smaller width of the wavefunction favors the solid phase over the liquid phase^{19}. This can explain an increased resistivity of HRP1, HRP2, and HRP3 phases and the gradual shift of insulating phase towards higher filling factors. In this scenario, the m_{CF} increase in ν < 1/2 region is consistent with our observation that m_{CF} is larger the closer the electron system approaches the high resistivity phase, which may be interpreted as an effective localization of the CF.
Finally, we discuss the filling factor ν = 1/2, which is originally thought to have a Fermi surface of CF (Fig. 1). Figure 4 depicts the temperature dependence of R_{xx} at several θ’s. At θ = 0, considering the presence of only CF Fermi surface, the logarithmic temperature dependence of R_{xx} points toward a residual CF interaction^{2,25,44}. At larger θ the slope of logarithmic R_{xx} temperature dependence increases, pointing to an increased CF residual interaction. On the other hand, our experiment demonstrates the shift of the insulating phase towards higher filling factors with an increasing θ. Consequently, the state at ν = 1/2 is formed by both a CF Fermi liquid with a Fermi surface and tail of the insulating phase forming in the course of liquidsolid phase transition. Then the R_{xx} temperature dependence at each θ in Fig. 4 reflects not only the CF residual interaction but also the melting of the intermediate liquidsolid phase growing towards ν = 1/2. Giving a small R_{xx} increase between θ = 0 and θ = 40 the intermediate phase is absent or its contribution is small at ν = 1/2 for θ = 0. Our consideration raises questions about the stability of a CF Fermi surface exposed to a strong inplane magnetic field, whereas the 2DES characteristics in a tilted magnetic field can likely be the attributes of the precursor for the new insulating state in a strong inplane magnetic field proposed by Piot et al.^{33}.
Our experimental data show that the electron system enters a correlation regime, which reflects the character of both solid and liquid phases for ν < 1/2 and can likely be the features of a nontrivial phase transition in the electron system occupying the lowest LL. Our experimental results are interpreted within the CF approach, which has recently attracted renewed attention from theory predicting that the CF can be Dirac particles^{45,46,47,48}. This also introduces an exciting perspective for ZnO studies. Our experimental results display the composite fermion paradigm in a system distinct from conventional semiconductor systems and address the question on how the charge carrier system translates between liquid and solid phases.
Methods
Sample
The sample under study is a MgZnO/ZnO heterostructure (a sequence of about 500nm thick epitaxial ZnO film (buffer layer) and 250nm thick MgZnO (capping layer) grown on ZnO substrate) with a charge carrier density n = 1.7 × 10^{11} cm^{−2} and a mobility μ = 600,000 cm^{2}V^{−1}s^{−1} at the base temperature of our dilution refrigerator T = 60 mK. The LL mixing for this charger carrier density is κ = 4.2 at ν = 1/2.
Magnetotransport in tilted magnetic field
The sample is mounted on a rotating stage allowing insitu sample rotation in the magnetic field. The tilt angle θ is determined from the shift of R_{xx} resistance minima of the wellknown fractional quantum Hall states.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
We acknowledge the support of the HFMLRU/FOM member of the European Magnetic Field Laboratory (EMFL). This work was partly supported by JST CREST No. JPMJCR16F1. We would like to thank M. Kawamura, A. S. Mishchenko, M. Ueda, K. von Klitzing, N. Nagaosa, and J. K. Jain for fruitful discussions and comments.
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D.M. and M.K. initiated the project. M.K. supervised the project. D.M. conceived and designed the experiment. J.F. and Y.K. fabricated the samples. D.M., A.M., J.B., and U.Z. performed the highfield magnetotransport experiment and analyzed the data. D.M. and U.Z. wrote the paper with input from all coauthors.
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Maryenko, D., McCollam, A., Falson, J. et al. Composite fermion liquid to Wigner solid transition in the lowest Landau level of zinc oxide. Nat Commun 9, 4356 (2018). https://doi.org/10.1038/s41467018068346
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